DPDM Experiential Learning : Data Visualization

Group 4

• Aman Garg : 21030242007
• Amandeep Sharma : 21030242008
• Jaikishan Patel : 21030242070
• Rut Patel : 21030242072

Dataset : US Police Shootings
Data Source : https://www.kaggle.com/ahsen1330/us-police-shootings

What is Data Visualization?

• Data visualization cares with the planning , development, and application of computer-generated graphical representation of the data.


• We can use different types of graphs, charts, plots etc which shows trends, patterns and relationship among the data. Data visualization is important aspect of data science.


• Common visualization techniques we see around us are mentioned below:
  1. Histogram
  2. Line Graph
  3. Bar Graph
  4. Pie Chart
  5. Box Plot
  6. Distplot
  7. Heatmap

Applications of Data Visualization

1. Healthcare industries: Data visualization will help to create a dashboard to visualize patients’ history.This can help doctor to analyse reports.


2. Finance: Finance is one of the most number driven. Data visualization can help in customer analysis and risk assessment


3. Manufacturing : We can use data visualization in analysing demand and supply of current stocks.

Advantages of Data visualizations

1. Human mind can understand and learn from visuals much faster and hence data visualization makes it easy to understand underlying trends in the data.


2. It can help understand certain situations quickly in order to take decisions faster.


3. People do not need to have a statistical or data science background to understand the graphical representations.

Some of the tools available in the market for data visualiztion

• Languages like R and Python have external libraries for data visualizations.
• Tableu
• Microsoft Power BI
• SAP Analytics Cloud
• Sisense

Graphs used our case study

1. Bar Graph : Bar graphs shows rectangular bars on an x-y plane that describe set of data points.


2. Line Graph : Group of data points are connected by line to showcase a trend over a period of time.


3. Pie Chart : A circular chart that divides it into proportions. Each sector can be represented as a percentage of a whole.


4. Box plot : Showcases distribution of data. Boxplots can also give a sense of outliers in the data. it also shows the median, Quartile-1, Quartile-3 and Inter quartile range.


5. Distplot : Showcases the distribution of a single variable against it's density.


6. Heatmap : Shows magnitude of a phenomenon as color in two dimensions

Data Visualization using Python

Libraries used:

1. Pandas
   • Provides data structures like Dataframe and Series.
   • Can read different file formats like excel, csv etc as dataframes.
   • Provides ability to process data easily.


2. Numpy
   • Short for Numerical Python
   • Provides arrays and matrix data structures which are much faster than list data structure python provides.
   • Provides many mathematical functions for processing.


3. Statsmodels
   • Provides users classes and functions that help users conduct statistical tests on any dataset.
   • Matplotlib is used for graphical representation for Statsmodels.


4. Scipy
   • Used for scientific and technical computing.
   • Contains modules for optimization, linear alegbra, image processing.


5. Matplotlib
   • Used for plotting 2D figures like graphs, histograms, scatter plots and more.
   • Uses Numpy and Pandas for plotting.


6. Seaborn
   • It is based on Matplotlib but generates more attractive figures.
   • Provides easier customization.


7. Plotly
   • It is another library for plotting figures.
   • Provides even more customization than Matlotlib and Seaborn : like hover tools.

# importing all the necessary libraries

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import plotly.express as px
import plotly.io as pio
import matplotlib
from scipy import stats
from statsmodels.stats.proportion import proportions_ztest
matplotlib.style.use('fivethirtyeight')
pio.templates.default = "plotly_dark"
plt.rcParams.update({'font.size': 18})

# Reading the dataset into a data frame
us_police_shootings=pd.read_csv("shootings.csv")

Preliminary data analysis

# Seeing at what the datasetlooks like 
us_police_shootings.head()

	id	name	date	manner_of_death	armed	age	gender	race	city	state	signs_of_mental_illness	threat_level	flee	body_camera	arms_category
0	3	Tim Elliot	2015-01-02	shot	gun	53.0	M	Asian	Shelton	WA	True	attack	Not fleeing	False	Guns
1	4	Lewis Lee Lembke	2015-01-02	shot	gun	47.0	M	White	Aloha	OR	False	attack	Not fleeing	False	Guns
2	5	John Paul Quintero	2015-01-03	shot and Tasered	unarmed	23.0	M	Hispanic	Wichita	KS	False	other	Not fleeing	False	Unarmed
3	8	Matthew Hoffman	2015-01-04	shot	toy weapon	32.0	M	White	San Francisco	CA	True	attack	Not fleeing	False	Other unusual objects
4	9	Michael Rodriguez	2015-01-04	shot	nail gun	39.0	M	Hispanic	Evans	CO	False	attack	Not fleeing	False	Piercing objects

# Checking number of rows and columns
us_police_shootings.shape

(4895, 15)

# Checking for null values in each column
us_police_shootings.isnull().sum()

id                         0
name                       0
date                       0
manner_of_death            0
armed                      0
age                        0
gender                     0
race                       0
city                       0
state                      0
signs_of_mental_illness    0
threat_level               0
flee                       0
body_camera                0
arms_category              0
dtype: int64

# Different columns in the data set
us_police_shootings.columns

Index(['id', 'name', 'date', 'manner_of_death', 'armed', 'age', 'gender',
       'race', 'city', 'state', 'signs_of_mental_illness', 'threat_level',
       'flee', 'body_camera', 'arms_category'],
      dtype='object')

# Unique values in the armed column
us_police_shootings['armed'].unique()

array(['gun', 'unarmed', 'toy weapon', 'nail gun', 'knife', 'unknown',
       'shovel', 'hammer', 'hatchet', 'sword', 'machete', 'box cutter',
       'metal object', 'screwdriver', 'lawn mower blade', 'flagpole',
       'guns and explosives', 'cordless drill', 'metal pole', 'Taser',
       'metal pipe', 'metal hand tool', 'blunt object', 'metal stick',
       'sharp object', 'meat cleaver', 'carjack', 'chain',
       "contractor's level", 'stapler', 'crossbow', 'bean-bag gun',
       'baseball bat and fireplace poker', 'straight edge razor',
       'gun and knife', 'ax', 'brick', 'baseball bat', 'hand torch',
       'chain saw', 'garden tool', 'scissors', 'pole', 'pick-axe',
       'flashlight', 'vehicle', 'spear', 'chair', 'pitchfork',
       'hatchet and gun', 'rock', 'piece of wood', 'bayonet', 'pipe',
       'glass shard', 'motorcycle', 'pepper spray', 'metal rake', 'baton',
       'crowbar', 'oar', 'machete and gun', 'air conditioner',
       'pole and knife', 'beer bottle', 'baseball bat and bottle',
       'fireworks', 'pen', 'chainsaw', 'gun and sword', 'gun and car',
       'pellet gun', 'BB gun', 'incendiary device', 'samurai sword',
       'bow and arrow', 'gun and vehicle', 'vehicle and gun', 'wrench',
       'walking stick', 'barstool', 'grenade', 'BB gun and vehicle',
       'wasp spray', 'air pistol', 'baseball bat and knife',
       'vehicle and machete', 'ice pick', 'car, knife and mace'],
      dtype=object)

# Unique values in the state column
us_police_shootings['state'].unique()

array(['WA', 'OR', 'KS', 'CA', 'CO', 'OK', 'AZ', 'IA', 'PA', 'TX', 'OH',
       'LA', 'MT', 'UT', 'AR', 'IL', 'NV', 'NM', 'MN', 'MO', 'VA', 'NJ',
       'IN', 'KY', 'MA', 'NH', 'FL', 'ID', 'MD', 'NE', 'MI', 'GA', 'TN',
       'NC', 'AK', 'NY', 'ME', 'AL', 'MS', 'WI', 'SC', 'DE', 'DC', 'WV',
       'HI', 'WY', 'ND', 'CT', 'SD', 'VT', 'RI'], dtype=object)

# Unique values in the arms_category column
us_police_shootings['arms_category'].unique()

array(['Guns', 'Unarmed', 'Other unusual objects', 'Piercing objects',
       'Sharp objects', 'Unknown', 'Blunt instruments', 'Multiple',
       'Electrical devices', 'Hand tools', 'Vehicles', 'Explosives'],
      dtype=object)

1. Top 10 weapons victims were armed with

us_police_shootings['count'] = 1
runs=us_police_shootings.groupby(['armed'])['count'].sum().reset_index()
runs.columns=['armed','total_shootings']
max_shootings=runs.groupby(['armed'])['total_shootings'].sum()

# Plotting the bar graph
plt.figure(figsize=(15,10))
max_shootings.sort_values(ascending=False)[:10].plot.bar()
plt.ylabel('Total shootings')
plt.xlabel('Type of weapon')
plt.title('Top 10 weapons')

Text(0.5, 1.0, 'Top 10 weapons')

2. Count of different races of the victims

# BAR GRAPH OF RACES OF THE SHOOTERS
plt.figure(figsize=(15,10))
sns.countplot(x="race", data=us_police_shootings)
plt.show()

3. How the victims died?

# MANNER OF DEATH GRAPH
manner_of_death = us_police_shootings['manner_of_death'].value_counts().reset_index().rename(
    columns={'index':'manner of death','manner_of_death':'count'})

plt.figure(figsize=(8,10))
ax = sns.barplot(x = 'manner of death', 
                 y = 'count', 
                 data = manner_of_death, 
                 palette = "Set2")
plt.title("Frequency vs Manner of Death")
plt.xlabel("Manner of Death")
plt.ylabel("Frequency")
plt.show()

4. Comparing victims by gender

#GENDER PIE CHART

us_police_shootings.groupby('gender').size().plot(kind='pie', 
                                                  autopct='%.2f',
                                                  ylabel='Gender',
                                                  figsize=(10,10))

<AxesSubplot:ylabel='Gender'>

5. Distribution of plice shootings by states

# PIECHART OF STATES OF SHOOTINGS

state_shooting=us_police_shootings["state"].value_counts()
fig=px.pie(state_shooting,
           values=state_shooting.values,
           names=state_shooting.index,
           title="Police shootings in US States",
           height=900)
fig.show()

6. 20 Cities with most shootings

city = us_police_shootings['city'].value_counts().reset_index().rename(
    columns={'index':'City', 'city':'Total Shootings'}).head(20)

fig = px.bar(city, x = 'City', 
             y = 'Total Shootings',
             text = 'Total Shootings',
             title='Top 20 Cities with most shootings', 
             height=800, 
             width=900)

fig.update_traces(marker_color=['pink','plum','mediumorchid', 'darkviolet','mediumpurple',
                                'purple','indigo','thistle','mediumvioletred','blueviolet',
                                'yellow','green','red','grey','gold','blue','orange',
                                'yellowgreen','indianred','cyan'])

fig.show()

7. Distribution of Gender and Age

medians = us_police_shootings.groupby('gender')['age'].median()
plt.figure(figsize=(15,5))
box = sns.boxplot(x=us_police_shootings['gender'], y=us_police_shootings['age'])
for i in range(len(medians)):
    box.annotate(str(medians[i]),xy = (i, medians[i]), horizontalalignment = 'center')
plt.title('Gender VS Age Distribution', fontsize=20)

Text(0.5, 1.0, 'Gender VS Age Distribution')

8. Frquency of arms category used

#ARMS CATEGORY USED DURING SHOOTING.

arms_category = us_police_shootings['arms_category'].value_counts().reset_index().rename(
    columns={'index':'Arms Category', 'arms_category':'Total'})
fig = px.bar(arms_category, 
             x = 'Arms Category', 
             y = 'Total',
             text = 'Total',
             title='Arms Category Used', 
             height=800, 
             width=900)

fig.update_traces(marker_color=['pink','lightgreen','gold','darkviolet','red','purple',
                                'orange','thistle','indianred','blueviolet'])
fig.show()

9. Mental illness check

count = us_police_shootings['signs_of_mental_illness'].value_counts()
pie_list=[count[False],count[True]]
mental_labels=["Mentally Stable","Not Mentally Stable"]

def absolute_value(val):
    return f'{val:.2f}%'

plt.figure(figsize=(10,10))
plt.pie(pie_list,labels=mental_labels, autopct=absolute_value)
plt.title("Mental Illness")
plt.legend()

<matplotlib.legend.Legend at 0x7f4245c18700>

10. Hypothesis

NULL: Attack threats is equal to 64% of all the threats Alternative:Attack threats is not equal to 64% of all the threats

us_police_shootings['threat_level']
count_threats=us_police_shootings['threat_level'].value_counts()
num_threats=count_threats['attack']+count_threats['other']+count_threats['undetermined']

# Perform one proportion z-test
proportions_ztest(count=count_threats['attack'],nobs=num_threats,value=.64)

(0.812740629941684, 0.4163667811506514)

To conclude p-value = 0.416366 > 0.05, we accept the null hypothesis and can say that attack threats are 64% of all the threats. For proof bar plot has been plotted below:

threat_values=[count_threats['attack'],count_threats['other'],count_threats['undetermined']]
threat_labels=us_police_shootings['threat_level'].unique()

plt.figure(figsize=(8,10))
plt.bar(threat_labels,threat_values,color='green',width=0.5)
plt.title("Threat Types")
plt.xlabel("Threats")
plt.ylabel("Number of Threats")

Text(0, 0.5, 'Number of Threats')

11. Number of men/women fleeing vs not fleeing

# Replacing all the values with a single value where car, foot and others are used as Fleeing
gender_flee= us_police_shootings[['gender','flee']]
gender_flee.loc[(gender_flee.flee != 'Not fleeing'),'flee']='fleeing'

# Creating a two way table 
cross_table = pd.crosstab(gender_flee['gender'],
                        gender_flee['flee'],margins = False)
cross_table.plot(kind='bar',
                 rot=0,color={'fleeing': "red", 'Not fleeing': "green"},
                 figsize=(10,7))

# Plotting the graph
plt.legend(['Not fleeing','Fleeing'],labelcolor='black')
plt.xlabel('Gender')
plt.ylabel('Numbers')
plt.title('Male Female Flee Chart', color='black')

/home/aman/Documents/my_env/lib/python3.8/site-packages/pandas/core/indexing.py:1637: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

/home/aman/Documents/my_env/lib/python3.8/site-packages/pandas/core/indexing.py:692: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

Text(0.5, 1.0, 'Male Female Flee Chart')

12. Body Camera off: All Races

us_police_shootings["body_camera"] = us_police_shootings["body_camera"].apply(str)
cam = us_police_shootings[["race","body_camera"]]
cam_off = cam[cam["body_camera"]=="False"]
cam_off = cam_off["race"].value_counts()
fig = px.pie(cam_off,
           values = cam_off.values,
           names = cam_off.index,
           title = "People belonging to Race who did not have Body Camera")
fig.show()

13. Race vs Age Distribution

# RACE VS AGE Boxplot
medians = us_police_shootings.groupby('race')['age'].median()
plt.figure(figsize=(15,10))
box = sns.boxplot(x=us_police_shootings['race'], y=us_police_shootings['age'])
for i in range(len(medians)):
    box.annotate(str(medians[i]),xy = (i, medians[i]), horizontalalignment = 'center')
plt.title('Race VS Age Distribution', fontsize=22)

Text(0.5, 1.0, 'Race VS Age Distribution')

14. Arms category: Other than guns

ms_cat=us_police_shootings[us_police_shootings["arms_category"]!="Guns"]
no_guns_plot=ms_cat["state"].value_counts()

plt.figure(figsize=(15,25))
sns.barplot(x=no_guns_plot,y=no_guns_plot.index)
plt.xlabel("Count")
plt.ylabel("States")
plt.title("Frequency of Objects other than guns in different States")
plt.plot()

[]

15. Number of shootings in each state where victim was armed with a gun.

victim_with_gun = us_police_shootings[us_police_shootings['arms_category'] == 'Guns'][['state','arms_category']].groupby('state').count().reset_index()
fig = px.bar(victim_with_gun,
       x = 'arms_category',
       y ='state',
       text="arms_category",
       color ='state',
       width = 970,
       height = 1500)
fig.show()

16. Relationship between mental illness and weapon use

# Replacing all the values with a single value where armed value is not unarmed
mental_health_weapon_use = us_police_shootings[['armed','signs_of_mental_illness']]
mental_health_weapon_use.loc[(mental_health_weapon_use.armed != 'unarmed'),'armed']='armed'

# Creating a two way table 
cross_table = pd.crosstab(mental_health_weapon_use['signs_of_mental_illness'],
                          mental_health_weapon_use['armed'],margins = False)
cross_table.plot(kind='bar',
                 rot=0,color={'armed': "red", 'unarmed': "green"},
                 figsize=(15,10))

# Plotting the graph
plt.legend(['Armed','Unarmed'],labelcolor='black')
plt.xlabel('Sign of mental illness')
plt.ylabel('Number of shootings')
plt.title('Relationship between armed and sign of mental illness', color='black')

/home/aman/Documents/my_env/lib/python3.8/site-packages/pandas/core/indexing.py:1720: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

Text(0.5, 1.0, 'Relationship between armed and sign of mental illness')

17. Trend in number of shooting over the years

us_police_shootings['year'] = pd.DatetimeIndex(us_police_shootings['date']).year
shootings_each_year = us_police_shootings['year'].value_counts().reset_index().rename(
    columns={'index':'year','year':'no of shootings'}).groupby('year').sum().reset_index()

# Plotting the graph
plt.figure(figsize=(10,10))
plt.plot(shootings_each_year['year'],
         shootings_each_year['no of shootings'],
         color='green', 
         marker='o',
         linestyle='solid')

[<matplotlib.lines.Line2D at 0x7f423fcac5b0>]

18. Finding Outliers by age

df=pd.read_csv('shootings.csv')
df['count'] = 1
df['year'] = pd.DatetimeIndex(df['date']).year
df['month'] = pd.DatetimeIndex(df['date']).month
age=df['age']

plt.figure(figsize =(10, 10))
plt.boxplot(age)
plt.xlabel("Plot")
plt.ylabel("Age")
box = sns.boxplot(data=age)
for i in range(len(medians)):
    box.annotate(str(medians[i]),xy = (i, medians[i]), horizontalalignment = 'center')

The above boxplot of age shows there are ouliers. Now, we will find outlier information.

median = np.median(age)
upper_quartile = np.percentile(age, 75)
lower_quartile = np.percentile(age, 25)

iqr = upper_quartile - lower_quartile
upper_whisker = age[age<=upper_quartile+1.5*iqr].max()
lower_whisker = age[age>=lower_quartile-1.5*iqr].min()
upper_whisker

72.0

People above 72 is considered outlier in box plot.

outlier_information=df[df['age']>72]
outlier_information['count'].count()

37

Since,The number of people is significant in number(37) whose age is cosidered outlier by box plot,we will consider them for analysis and will not delete them.

19. Checking the distribution age follows

plt.figure(figsize=(15,10))
sns.set_style("darkgrid")
sns.distplot(df['age'],kde=True,color='b')

/home/aman/Documents/my_env/lib/python3.8/site-packages/seaborn/distributions.py:2557: FutureWarning:

`distplot` is a deprecated function and will be removed in a future version. Please adapt your code to use either `displot` (a figure-level function with similar flexibility) or `histplot` (an axes-level function for histograms).

<AxesSubplot:xlabel='age', ylabel='Density'>

From Displot, we can see it it is very slightly positively skewed. But, We can confidently say it is following normal distribution.

plt.figure(figsize=(15,10))
stats.probplot(df['age'],plot=plt)

((array([-3.63021417, -3.39456193, -3.26466538, ...,  3.26466538,
          3.39456193,  3.63021417]),
  array([ 6.,  6., 12., ..., 84., 84., 91.])),
 (12.449614166604643, 36.549749566213784, 0.980154295559862))

From Probplot, Majority of data is following line. So, We can say it is following normal distribution.

Shapiro Test

The Shapiro-Wilk’s test or Shapiro test is a normality test in frequentist statistics. The null hypothesis of Shapiro’s test is that the population is distributed normally.

stats.shapiro(df['age'])

ShapiroResult(statistic=0.9606190323829651, pvalue=1.6336742685352274e-34)

By abpplying shapiro test in order to check whether age is following normal distribution or not. We came to know that p values is equal to 0 that signifies data follows normal distribution.

20. Monthly police shootings from 2015-2020

year_month_totalshootings=df.groupby(['year','month'])['count'].sum().reset_index()
year_month_totalshootings.columns=['year','month','total_shootings']
heatmap_year_month_totalshootings=year_month_totalshootings.pivot('month','year','total_shootings')
plt.figure(figsize=(15,10))
sns.heatmap(heatmap_year_month_totalshootings,annot=True)

<AxesSubplot:xlabel='year', ylabel='month'>

Top three months when more number of people were shot is 01-2019,01-2018 and 12-2019 respectively.

21. Age Range vs Mental Illness

age_range=["[{0} - {1})".format(age,age+10)for age in range(0,101,10)]
count_age_ranges=len(age_range)
df['age_range']=pd.cut(x=df['age'],bins=count_age_ranges,labels=age_range)
agerange_totalshootings=df.groupby(['age_range'])['count'].sum().reset_index()
agerange_totalshootings.columns=['age_range','total_shootings']
agerange_totalshootings# .plot.bar()

##
mental=df[df['signs_of_mental_illness']==True]
agerange_totalshootings_mental=mental.groupby(['age_range'])['count'].sum().reset_index()
agerange_totalshootings_mental.columns=['age_range','total_shootings']
agerange_totalshootings_mental
agerange_totalshootings['mentally disable']=agerange_totalshootings_mental['total_shootings']
agerange_totalshootings['percentage of mentally disable']=(agerange_totalshootings['mentally disable']/agerange_totalshootings['total_shootings'])*100
agerange_totalshootings['percentage of not mentally disable']=100-agerange_totalshootings['percentage of mentally disable']

##
plt.figure(figsize=(16, 7))
plt.bar(agerange_totalshootings['age_range'],agerange_totalshootings['percentage of mentally disable'] , color='g')
plt.bar(agerange_totalshootings['age_range'], agerange_totalshootings['percentage of not mentally disable'],bottom=agerange_totalshootings['percentage of mentally disable'] ,color='y')
plt.xlabel("Age bracket")
plt.ylabel("Percentage of mentally ill and well")
plt.show()

in age bracket of '70 to 80' and '100-110',more people was mentally ill as compared to other age brackets.We can say that by increase in age,percentage of mentally ill people also increases.

22. States where more percentage of police shooting's victims were under adult.

age_below_maturityage=df.groupby(['state'])['age'].quantile(0.05).reset_index()
age_below_maturityage.columns=['state','age']
age_below_maturityage.sort_values('age').head(5)

	state	age
14	IL	17.0
2	AR	17.6
6	CT	17.9
22	MI	18.0
21	ME	18.0

CONCLUSIONS

1. Majority of People who were shot down and having weapon, had guns and knives respectively.


2. The people were shot down was mainly of white race,Black race and Hispanic. Asian,Native and others are significantly lesser in number as compared to top three.


3. Number of males was 21 times the number of females who were shot down.


4. Higest number of police shootings can be seen CA,TX and FL respectively. Combining number of all three states is more than 29% of total shootings.


5. Median age for both men and women is the same either 35 years


6. As expected from state's data,largest city of california Los angles had highest shooting. largest cities of Arizona and Texas are Phoenix and Houstan respectively also reported more number of police shootings than other cities.


7. Guns are the most used weapons which is expected given the gun laws in the US.


8. 77.47% of the total cases were mentally stable, the remaining 22.53% were not mentally stable.


9. The median ages of every race are lying under age bracket of 30 to 38.White race's boxplot clearly depicts that distribution of age is significant than other races.


10. From the graphs we see there is no significant relation between mental illness and using a weapon.


11. We see a decreasing trend in number of shootings each year and the dip in the year 2020 is due to lack of data.


12. California and Texas are the top states where the victims are armed with gun.


13. In IL, AR,CT, MI and ME, more than 5% of respective population of state were not adults.

Each Memeber's Contribution

• Aman Garg : 13-17, Theory, Conclusions


• Amandeep Sharma : 8-12, Theory, Conclusions


• Jaikishan Patel : 1-6, Theory, Conclusions


• Rut Patel : 18-22, Theory, Conclusions